In this paper, the four integral identities satisfied by the fundamental solution for elastostatic problems are reviewed and slightly different forms of the third and fourth identities are presented. Two new identities, namely the fifth and sixth identities, are derived. These integral identities can be used to develop weakly singular and nonsingular forms of the boundary integral equations (BIEs) for elastostatic problems. They can also be employed to show the nonuniqueness of the solution of the traction (hypersingular) BIE for an elastic body on a multiconnected domain. This nonuniqueness is shown in a general setting in this paper. It is shown that the displacement (singular) BIE does not allow any rigid-body displacement terms, while the traction BIE can have arbitrary rigid-body translation and rotation terms, in the BIE solutions on the edge of a hole or surface of a void. Therefore, the displacement solution from the traction BIE is not unique. A remedy to this nonuniqueness solution problem with the traction BIE is proposed by adopting a dual BIE formulation for problems with multiconnected domains. A few numerical examples using the 2D elastostatic boundary element method for domains with holes are presented to demonstrate the uniqueness properties of the displacement, traction and the dual BIE solutions for multiconnected domain problems.